Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with $$ F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X) $$ where the right hand side denotes arbitrary maps of sets (of course it wouldn't make sense to say order preserving as $X$ doesn't come with an order).

I'm wondering about a description of $F^{\bullet}(X)$. For example if $X = \{0,1\}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.

**Edit**: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.

Is there an analogous description when $X = \{0, 1, 2\}$?

A closely related question is whether there's a right adjoint to the forgetful functor from the simplex category $\Delta$ (finite ordered sets) to, say, finite (unordered) sets -- and if so what is it.

*Example where such simplicial sets arise*: given a map of topological spaces $f: X
\to Y$ we can always form a
simplicial object $\mathcal{S}^{\bullet}(f)$ with
$$
\mathcal{S}^{n} = \prod\nolimits_{X}^{n} = \underbrace{X \times_{Y}
\cdots \times_{Y} X}_{n\text{ times }}
$$
with face and degeneracy maps given by projections and diagonals
respectively. Taking connected components gives a simplicial set.

When $Y$ is the union $\bigcup_{i=1}^{N} H_{i}$ of the coordinate hyperplanes in $\mathbb{C}^{N}$ and $f: X=\coprod_{i=1}^{N} H_{i} \to \bigcup_{i=1}^{N} H_{i}=Y$ is the obvious map, I believe the simplicial set we get is $F^{\bullet}(\{1, \dots, n\})$.